Union of $C^*$-algebras generated by one element also generated by one element?

Question

Let $A$ be a unital $C^*$-algebra and $a_1, a_2, ... \in A$ be elements such that $C^*(a_1, 1) \subseteq C^*(a_2, 1)\subseteq ...$ where $C^*(., 1)$ denotes the generated unital $C^*$-algebra. The closure of the union $B:=\overline{\bigcup_{i=1}^\infty C^*(a_i, 1)} \subseteq A$ again is a $C^*$-algebra. My question: Is $B$ also of the form $B=C^*(a, 1)$ for a suitable element $b\in A$ or is there an element $b\in A$ such that $B\subseteq C^*(b, 1)$?